Download Introduction to symmetric spectra I

INTRODUCTION TO SYMMETRIC SPECTRA.
LECTURE 1
In this lecture we will concentrate on the algebraic (rather than homotopy-theoretic)
aspects of the theory of symmetric spectra. In particular, the discussion of model categories will be postponed until the next lecture, and the highlight of this talk will be
the construction of a closed symmetric monoidal structure on the category of symmetric
spectra. (The existence of such a structure is one of the main technical advantages of the
theory of symmetric spectra).
The main reference we have used in preparing these notes is
[HSS] M. Hovey, B. Shipley, and J. Smith, “Symmetric spectra”, Journal of the AMS
13 (1999), no. 1, 149 – 208.
A very good background reference for this lecture and the next one is
[GM] S. Gelfand and Yu. Manin, “Methods of Homological algebra”
In particular, [GM] contains a detailed discussion of simplicial sets and model categories.
Two of the more classical references on simplicial sets and homotopy theory are
[GZ] P. Gabriel and M. Zisman, “Calculus of fractions and homotopy theory”
[M] P. May, “Simplicial objects in algebraic topology”
1. Simplicial sets. The goal of this section is mostly to fix the notation. Let
4
4big denote the category of finite linearly ordered sets and nondecreasing maps, and let
4
4 ⊂ 4
4big be the full subcategory formed by the objects
[n] = {0, 1, 2, . . . , n},
n ∈ Z, n ≥ 0.
op
Definition: If C is a category, a simplicial object in C is a functor X : 4
4
→ C.
Thus, for any n ∈ Z≥0 , we have an object
X(n) := X([n]) ∈ Ob(C),
and for every nondecreasing map f : [m] → [n], we have a morphism
X(f ) : X(n) → X(m)
in C, such that X(g ◦ f ) = X(f ) ◦ X(g) whenever g ◦ f is defined. Of course, simplicial
op
objects in C form a category, Funct(4
4 , C). As we have seen in the previous lecture
(on the K-theory of Waldhausen categories), sometimes it is also convenient to consider
Q-simplicial objects (which generalize simplicial objects, bisimplicial objects, etc.).
1
Q
Q
Definition: Let Q be a finite set. We write 4
4 ⊂ 4
4big for the products of copies of
the categories 4
4 and 4
4big indexed by Q. If the reader prefers, she/he may think of Q as
a discrete category (whose set of objects is Q and which has no non-identity morphisms)
and define
Q
Q
4
4 = Funct(Q, 4
4),
4
4big = Funct (Q, 4
4big ) .
If C is any category, we define the category of Q-simplicial objects of C as
Q op
,C .
S Q C := Funct 4
4
Q op
The diagonal of a Q-simplicial object X : 4
4
→ C is the simplicial object
diag(X) :
op
op
Q op
4
4
→ C obtained by composing X with the diagonal inclusion 4
4
,→ 4
4
.
Remark: The following comment was used implicitly in the previous lecture. Let
ΣQ = Aut(Q) denote the group of permutations of the set Q (if |Q| = m, then ΣQ ∼
= Σm ,
Q
the symmetric group on m letters). Then ΣQ acts on 4
4 by permuting the factors, and
hence there is an obvious
notion
of
a
Σ
-equivariant
Q-simplicial object of C, namely,
Q
Q op
→ C equipped with a compatible collection of isomorphisms
it is a functor X : 4
4
'
∗
of functors σ X −→ X for all σ ∈ ΣQ . In this situation, the diagonal diag(X) inherits
an action of ΣQ . (This is what was used in the construction of the K-theory of a Waldhausen category as a symmetric spectrum that we have presented.) We leave the precise
formulation of this fact to the reader.
The main example of simplicial objects for us is the category of simplicial sets,
i.e., simplicial objects in the category of sets. We denote the category of simplicial sets
simply by S. In what follows it will be convenient to use the language introduced by
Grothendieck; thus a simplicial set is the same thing as a presheaf of sets on the category
4
4. When no confusion is possible, we will omit the word “sets” and simply speak of a
“presheaf on 4
4”. By abstract nonsense, the category S is complete and cocomplete. For
each n ∈ Z≥0 , we have the representable presheaf 4[n] ∈ S, defined by
4[n](m) = 4
4([m], [n]).
(Here we use the convention that if C is a category and X, Y are objects of C, then
C(X, Y ) denotes the set of morphisms X → Y in C.) By Yoneda’s lemma, we then have,
for any X ∈ S,
S(4[n], X) ∼
= X(n) canonically.
In particular, let ∗ = 4[0]; this is a simplicial set with ∗(m) consisting of one point for
each m ∈ Z≥0 , and it is clear that ∗ is a terminal object in S.
An initial object, ∅, also exists in S (though it is not representable); it is defined by
∅(m) = ∅ for all m ∈ Z≥0 .
The category of pointed simplicial sets (i.e., the category of arrows ∗ → X in S)
is denoted by S∗ ; note that it is naturally equivalent to the category of simplicial objects
in the category of pointed sets.
2. Geometric realization. Arguably, the notion of geometric realization could
be “discovered” by a beginner in the following way. It is known that for a general category
D, every functor Dop → Sets is a colimit of representable ones. Thus if we require the
geometric realization functor
| · | : S −→ Top = category of topological spaces
2
to commute with colimits and satisfy the (obviously very natural) property
n
o
X
n+1 (∗)
|4[n]| = (x0 , . . . , xn ) ∈ R
xi = 1 ,
xi ≥ 0,
then the geometric realization of any simplicial set would be uniquely determined.
Let us give some details. For the abstract categorical aspect, we refer the reader to
op
[GZ], § II.1.2. Fix a simplicial set X : 4
4
→ Sets. For each n ≥ 0, consider the simplicial
set X(n) × 4[n] (i.e., the disjoint union = coproduct of copies of 4[n] indexed by X(n) ),
and form the disjoint union
a
P 0 :=
X(n) × 4[n] ∈ S.
n≥0
Next, for each arrow f : [m] → [n] in 4
4, let us write d(f ) = m and c(f ) = n (here “d”
stands for “domain” and “c” for “codomain”) and form
a
P 1 :=
X(c(f )) × 4[d(f )] ∈ S.
f ∈Ar 4
4
There are two natural morphisms
α
P1
β
/
/ P0 ,
namely, α is induced by the morphisms
X(f )×id
X(c(f )) × 4[d(f )] −−−−−→ X(d(f )) × 4[d(f )],
and β is induced by the morphisms
id ×4[f ]
X(c(f )) × 4[d(f )] −−−−−→ X(c(f )) × 4[c(f )]
for all f ∈ Ar 4
4, where
4[f ] : 4[m] → 4[n]
is the morphism induced by f : [m] → [n].
We also have a natural morphism
γ
P 0 −→ X
induced by the compositions
ev
X(n) × 4[n] ∼
= S(4[n], X) × 4[n] −→ X
(the last arrow is the obvious “evaluation map”). The following result is very important;
its proof is left as an easy but instructive exercise.
Theorem: The diagram P 1
α
β
/
/P 0
γ
/ X is a coequalizer in S.
Remark: The reason for discussing this basic material in such detail is that its understanding is necessary for the later understanding of why the symmetric monoidal structures on pointed simplicial sets, symmetric spectra, etc., are closed.
3
The discussion above motivates the following
Definition:∗) For each n ∈ Zn≥0 , define the standard (topological) n-simplex
|4[n]| by the formula (∗). If X is a simplicial set, its geometric realization |X| is
defined as the coequalizer of the pair
`
(∗∗)
X(c(f )) × |4[d(f )]|
f ∈Ar 4
4
|α|
|β|
/
/ `
X(n) × |4[n]| ,
n≥0
where |α| and |β| are defined in a way completely analogous to the one above, and we
only point out that for an arrow f : [m] → [n] in 4
4, we let
|4[f ]| : |4[m]| → |4[n]|
be the unique affine map of simplices which on the vertices coincides with f .
Remarks: (1) The definition of |X| is obviously functorial. For any morphism ϕ :
X → Y of simplicial sets, the corresponding continuous map of the geometric realizations
is denoted by |ϕ|.
(2) The definition of |X| given above works equally well for simplicial topological
spaces, i.e., simplicial objects in Top. Namely, in (∗∗), one needs to view X(c(f )) and
X(n) as topological spaces, and the product has to be taken in Top; everything else goes
through in the same way.
(3) There is an alternate description of |X|, which is easily seen
to the
to be equivalent
`
one given above. Namely, |X| can be defined as the quotient
X(n) × |4[n]|
∼,
n≥0
where ∼ is the equivalence relation generated by the relation
(f (x), p) ≈ (x, |4[f ]|(p))
f
for all [m] → [n], x ∈ X(n) , p ∈ |4[m]|. (The quotient is equipped with the quotient
topology.) This description is probably the most classical one.
(4) For some purposes the definition of geometric realization we gave is not the best
one. A more “modern” definition is obtained by extending a given X ∈ S to a functor
op
4
4big → Sets (such an extension is unique, since objects of 4
4big have no nontrivial automorphisms), and defining the geometric realization of such a functor in a “coordinate-free”
way. See: [V. Drinfeld, “On the notion of geometric realization”].
3. Smash products in S∗ . Since the categories Sets and Sets∗ (the latter
is that of pointed sets) are complete and cocomplete, so are the categories S and S∗ (by
abstract nonsense). Two special cases of this statement deserve explicit mention.
First, we have quotients. If j : Y → X is a morphism in S (which the reader may
assume to be injective), the quotient X/Y is defined as the pushout
Y
j
/X
/ X/Y
∗
Note that by construction, X/Y is naturally defined as an object of S∗ . This makes sense,
in particular, when Y = ∅; then we write X+ := X/∅ ∼
= X t ∗. This defines a functor
S → S∗ ,
X 7→ X+ ,
4
which is left adjoint to the forgetful functor S∗ → S.
Next we have products. The categorical product of two objects X, Y ∈ S is defined
by X × Y ; explicitly,
(X × Y )(n) = X(n) × Y(n)
∀n ≥ 0.
We also introduce the smash product in S∗ , which is different from the categorical
product in S∗ ; if X, Y ∈ S∗ , we define
X ∧ Y = (X × Y )/((X × ∗) ∪ (∗ × Y )).
Theorem: The smash product defines a closed symmetric monoidal structure on S∗ .
The fact that it is a symmetric monoidal structure is a straightforward exercise. For
the fact that it is closed, we recall first that it means the following: for any X, Y ∈ S∗
there exists Hom(X, Y ) ∈ S∗ and a functorial isomorphism
S∗ (Z ∧ X, Y ) ∼
= S∗ (Z, Hom(X, Y ))
(†)
for all Z ∈ S∗ . To prove the existence of this inner hom, we observe first that if we take
Z = 4[n]+ for some n ≥ 0, then
Hom(X, Y )(n) ∼
= S(4[n], Hom(X, Y ))
∼
= S∗ (4[n]+ , Hom(X, Y ))
∼
= S∗ (4[n]+ ∧ X, Y ),
(Yoneda)
(adjunction)
(by (†))
which already determines Hom(X, Y ). Conversely, if we define Hom(X, Y ) by
Hom(X, Y )(n) := S∗ (4[n]+ ∧ X, Y )
for all n ≥ 0, then one can check that (†) will be satisfied for all z ∈ S∗ , not just for the
representable ones.
Remark: This construction of inner hom is really a special case of the following
important general fact. A functor
PSh(D)op → Sets,
where D is an arbitrary category and PSh(D) = Funct(Dop , Sets), is representable if and
only if it transforms colimits into limits. The “only if” direction is true (more or less by
definition) when PSh(D)is replaced by any category, but the “if” direction is a special
property of the category of presheaves.
4. Examples of simplicial sets.
1) We define the simplicial spheres as follows. First of all, by convention,
S 0 := 4[0]+ = ∗ t ∗ ∈ S∗ .
The simplicial circle is defined by
S∗ 3 S 1 := 4[1]/∂4[1] = coequalizer(4[0] ⇒ 4[1]),
where the two maps 4[0] ⇒ 4[1] are induced by the only two maps [0] ⇒ [1]. More
generally, for each n ≥ 2, we define
S∗ 3 S n := S 1 ∧ S 1 ∧ . . . ∧ S 1
5
(n copies).
Note that this is NOT isomorphic to the more classical version of S n defined as ∂4[n + 1],
i.e., 4[n + 1] with the interior removed. The advantages of the definition above are that
S n has a distinguished base point and comes with a natural action of the symmetric group
Σn . This will be important when we define symmetric spectra.
2) If X is a topological space, we define the total singular complex Sing(X) by
Sing(X)(n) = Top(|4[n]|, X)
∀n ≥ 0,
f
Sing(X)f = |4[f ]|∗
∀[m] → [n].
The functor Sing : Top → S is right adjoint to the geometric realization functor. We refer
the reader to [M], Ch. III, for details.
3) If C is a small category, its nerve N• C is the simplicial set defined by
N(n) C = Funct([n], C)
∀n ≥ 0
(recall that [n] can be viewed as a category). In fact, if Cat is the category of small
categories, then
N• : Cat → S
is a fully faithful functor ([GZ], § II.4.3).
More generally, if C is a Q-simplicial category, i.e., a Q-simplicial object in Cat, it
is clear that N• C is a (Q+1)-simplicial set. We write BC = |N• C| and call it the
classifying space of C.
5. Spectra and symmetric spectra.
Def 1: A spectrum E is given by the data of:
(i) a sequence E0 , E1 , . . . , En , . . . of pointed simplicial sets, and
(ii) a sequence of (pointed) morphisms
σ : S 1 ∧ En → En+1
∀n ≥ 0,
called the structure maps of the spectrum. The category of spectra is denoted SpN .
Remarks. (1) We could take the En ’s to be pointed topological spaces, or pointed
CW complexes, etc. This gives the original definition of a spectrum due to Lima and
Whitehead. Note also that some people call this a prespectrum, and reserve the name
“spectrum” for the situation where the structure maps are monomorphisms (i.e., cofibrations with respect to an appropriate model category structure on pointed simplicial sets).
There are also other versions.
(2) If E is a spectrum, then for each k ∈ Z, we can define the k-th homotopy group
of E as
πk (E) :=
lim πk+n (En ),
−→
n≥max{0,k}
where the colimit is taken with respect to the compositions
susp.
πk+n (En ) −−−→ πk+n+1 (S 1 ∧ En ) −−−→ πk+n+1 (En+1 ).
%
%
induced by S k+n+1 ∼
=S 1 ∧S k+n
induced by σ
We say that a morphism f : E → E 0 of spectra is a stable homotopy equivalence if
the homomorphism πk (f ) : πk (E) → πk (E 0 ) is an isomorphism for all k ∈ Z. Inverting
6
all stable homotopy equivalences in SpN , we obtain a certain category called the stable
homotopy category. It is equivalent to the s.h. category originally introduced by
Boardman; his construction was written up by Vogt [Vo70].
This category has been extensively studied by algebraic topologists, and as far as I
understand, there is no doubt that it is the right category to look at after inverting stable
homotopy equivalence. In particular, each of the many different known constructions of
categories of spectra all have the property that the associated stable homotopy category
is equivalent to the one introduced by Boardman.
(3) However, there is a problem with the definition of the category of spectra given
above. An important property of Boardman’s stable homotopy category is that it has a
symmetric monoidal structure; however, it is not induced from any such structure on the
category of spectra. All other categories of spectra defined in a similar manner share this
disadvantage. This has caused algebraic topologists a lot of frustration: for example, the
absence of a monoidal structure means that one does not have a good notion of a ring
spectrum. (This problem was solved in some sense by Peter May and collaborators, who
introduced the notion of an E∞ ring spectrum.)
In the late 1990s / early 2000s at least three solutions of this problem were proposed.
Two of them were developed by Peter May and collaborators, resulting in the theories of
“S-modules” and “orthogonal spectra”. The third one was suggested in a work of Hovey,
Shipley, and Smith; it results in the category of symmetric spectra and is one of the main
subjects of the present series of talks.
One of the main advantages of the category of symmetric spectra is that it has a
“smash product” which is associative and symmetric “on the nose”. On the other hand,
the theory of symmetric spectra is technically much simpler than the theory of S-modules,
yet it suffices for all our
applications, which is why we have chosen to work with it.∗∗)
Def 2: A symmetric spectrum is a spectrum E in the sense of Definition 1, equipped
with the additional data of the action of the symmetric group Σn on En by automorphisms
for each n ≥ 0, such that ∀p ≥ 1, the composition
/ Ep+n
O
S p ∧ En
σ
(S 1 ∧ . . . ∧ S 1 ) ∧ En
/ (S 1 ∧ . . . ∧ S 1 ) ∧ E
n+1
/ ...
/ S 1 ∧ En+p−1
induced by the structure maps of the spectrum is Σp × Σn -equivariant, where the action
of Σp × Σn on En+p is induced by the obvious injection Σp × Σn ,→ Σp+n .
A morphism f : E → F of symmetric spectra is a collection of Σn -equivariant
morphisms fn : En → Fn in S∗ ∀n ≥ 0, which commute with the structure maps in the
obvious sense. The category of symmetric spectra is denoted by SpΣ , and the forgetful
functor is denoted by
U : SpΣ → SpN
(here U stands for “underlying spectrum”).
Remark: With the correct notion of stable homotopy equivalence of symmetric spectra, a map f : E → F such that U(f ) : U(E) → U(F ) is an s.h.e. is itself an s.h.e., but
not vice versa!
7
Example: If K is a pointed simplicial set, its symmetric suspension spectrum
Σ∞ K is the symmetric spectrum defined by
(Σ∞ K)n = S n ∧ K,
where the Σn -action comes from its natural action on S n , and the structure maps
σ : S 1 ∧ S n ∧ K → S n+1 ∧ K
are induced by the obvious isomorphisms
S1 ∧ Sn ∼
= S n+1 .
In particular, the symmetric sphere spectrum is defined by S = Σ∞ S 0 =
(S 0 , S 1 , S 2 , . . .).
6. Simplicial structure on SpΣ .
If E is a symmetric spectrum and K is a pointed simplicial set, then E ∧ K is defined
in the obvious way: (E ∧ K)n = En ∧ K, where Σn acts trivially on K, and the structure
maps are σnew = σold ∧ idK .
On the other hand, if E and F are symmetric spectra, we define the simplicial set
MapSpΣ (E, F ) of maps from E to F by
MapSpΣ (E, F )(n) = SpΣ (E ∧ 4[n]+ , F ).
Of course, the definition models the construction of inner hom on the category of simplicial
sets. With this notation, we have the following
Theorem: (1) The assignment
K
(E 7→ E ∧ K)
defines a right action of S∗ (as a monoidal category) on SpΣ .
(2) We have bifunctorial isomorphisms
SpΣ (E ∧ K, F ) ∼
= S∗ (K, MapSpΣ (E, F ))
for all E, F ∈ SpΣ and all K ∈ S∗ . In particular, we have the evaluation map
evE,F : E ∧ MapSpΣ (E, F ) → F,
defined as the adjoint of
id : MapSpΣ (E, F ) → MapSpΣ (E, F ).
(3) The composition pairing
MapSpΣ (X, Y ) ∧ MapSpΣ (Y, Z) → MapSpΣ (X, Z),
defined as the adjoint of the composition
evX,Y ∧ id
evY,Z
X ∧ Map(X, Y ) ∧ Map(Y, Z) −−−−−−
→ Y ∧ Map(Y, Z) −−−→ Z,
8
is associative, and the adjoint S 0 → MapSpΣ (E, E) of the isomorphism E ∧ S 0 → E is a
unit for the composition. Hence SpΣ is naturally enriched over S∗ .
(For more information, see [HSS], § 1.3.)
In other words, SpΣ is a simplicial category with a closed right action of S∗ . The rest
of the lecture is devoted to a construction of a closed symmetric monoidal structure on
SpΣ .
7. Symmetric sequences. Before defining smash product on SpΣ , it is convenient to define it on a larger category (here, “larger” is understood in the sense that
there are more objects as well as more morphisms).
Definition: Consider the groupoid Σ whose objects are the finite sets n = {1, 2, . . . , n}
for all n ≥ 0 (with the convention that 0 = ∅) with morphisms
Σ(m, n) = ∅ for m 6= n,
and Σ(n, n) = Σn for all n ≥ 0.
It is (non-uniquely) equivalent to the groupoid of finite sets and bijections.
If C is a category, the category of symmetric sequences in C is defined as
CΣ = Funct(Σ, C).
In more down-to-earth terms, a symmetric sequence in C is a sequence {Xn }n≥0 of objects
of C, together with an action of Σn on each Xn (with no compatibility conditions).
In particular, we have the category S∗Σ of symmetric sequences of pointed simplicial
sets. By definition, there is a faithful (but certainly not full!) forgetful functor
SpΣ → S∗Σ .
Note, on the other hand, that the category of symmetric sequences is technically much
simpler than the category of symmetric spectra, due to the formal computation
S∗Σ = Funct(Σ, S∗ )
op
= Funct(Σ, Funct(4
4 , Sets∗ ))
op
∼
4 , Sets∗ )
= Funct(Σ × 4
op
∼
4) , Sets∗ ).
= Funct((Σop × 4
Thus, the category of symmetric sequences (hereafter, we omit the words “of pointed
simplicial sets” when no confusion can arise) is equivalent to the category of pointed
op
presheaves of sets on Σop × 4
4. (Footnote: in fact, Σ
' Σ by elementary group theory.)
In particular, the formalism of presheaves applies equally well to this category.
Remark: This is not stated explicitly in [HSS], and for this reason § 2.1 in that paper
is somewhat hard to read.
Definition: The tensor product of two symmetric sequences X, Y ∈ S∗Σ is the
symmetric sequence X ⊗ Y defined by
_
^
(X ⊗ Y )n =
(Σn )+
(Xp ∧ Yq ).
Σp ×Σq
p+q=n
9
Let usW
explain the notation.
1)
denotes the wedge operation (= coproduct)Win the category S∗ . If {Xi }i∈I is
any collection of pointed simplicial sets, their wedge
Xi is obtained from the disjoint
i∈I
`
union
Xi by identifying all the base points to a single base point.
i∈I
2) Σn can be viewed as a constant simplicial set (= constant presheaf on 4
4) defined
by
for all m ≥ 0,
for all f ∈ Ar 4
4.
(Σn )(m) = Σn
(Σn )f = idΣn
Then (Σn )+ is the pointed simplicial set associated to Σn .
3) The group Σp × Σq acts on Σp+q by right translations, and hence it also acts on the
pointed simplicial set (Σp+q )+ .
4) If K and L are pointed simplicial sets and G is a discrete group acting on K on
the right and on L on the left, we define K ∧ L as the quotient of K ∧ L by the diagonal
G
action of G given by
g : (k, l) 7→ (kg −1 , gl).
Thus (Σn )+
V
(Xp ∧ Yq ) is now defined, and moreover inherits a left action of Σn from
Σp ×Σq
its action on (Σn )+ . Hence X ⊗ Y is indeed a symmetric sequence.
Remark: The construction of
(Σn )+
^
(Xp ∧ Yq )
Σp ×Σq
is reminiscent of the construction of an induced representation.
Theorem: The tensor product defines a closed symmetric monoidal structure on S∗Σ .
For the proof of this fact we refer the reader to [HSS], § 2.1. As remarked above, the
existence of inner homs is essentially a formal statement. Note also that inner homs can
be constructed in two steps. First, we have a right action of S∗ on S∗Σ defined by
X ∈ S∗Σ , K ∈ S∗
X ∧ K ∈ S∗Σ ,
where (X ∧ K)n = Xn ∧ K with the diagonal action of Σn , induced by the
given action on Xn and the trivial action on K.
This action is closed, i.e., for X, Y ∈ S∗Σ we have a “simplicial set of maps” MapS∗Σ (X, Y )
such that there are functorial isomorphisms
S∗Σ (X ∧ K, Y ) ∼
= S∗ K, MapS∗Σ (X, Y )
for all X, Y ∈ S∗Σ and K ∈ S∗ .
Thus S∗Σ also becomes a simplicial category. The inner hom can then be constructed using
this simplicial structure; we note that if X, Y ∈ S∗Σ , then HomS∗Σ (X, Y ) is a symmetric
sequence whose 0-th term is MapS∗Σ (X, Y ). We will also describe the braiding on S∗Σ (and
'
omit all further details). The symmetry isomorphism X ⊗ Y −→ Y ⊗ X is induced from
isomorphisms
'
(Σp+q )+ ∧ Xp ∧ Yq −→ (Σp+q )+ ∧ Yq ∧ Xp
(†
†)
10
that commute with the diagonal action of Σp × Σq ∼
= Σq × Σp and thereby descend to
^
^
'
(Σp+q )+
(Xp ∧ Yq ) −→ (Σp+q )+
(Yq ∧ Xp ).
Σp ×Σq
Σq ×Σp
A moment’s thought will convince the reader that the only natural way of defining an
isomorphism (†
†) with this property is
(α, x, y) 7→ (α · ρq,p , y, x),
where ρq,p ∈ Σp+q is the (q, p)-shuffle, defined by
(
i + p if 1 ≤ i ≤ q,
ρq,p (i) =
i − q if q + 1 ≤ i ≤ q + p.
The proof that the isomorphism X ⊗ Y ∼
†) is a braiding on S∗Σ whose
= Y ⊗ X induced by (†
square is 1 is completely straightforward.
Remark: It is easy to verify that the unit object for the monoidal structure on S∗Σ is
the symmetric sequence (S 0 , ∗, ∗, ∗, . . .).
8. Smash products in SpΣ . The construction is based on the following
fundamental observations. 1) Let S 0 ∈ S∗Σ denote the sphere sequence, defined by
(S 0 )n = S n = S 1 ∧ . . . ∧ S 1
(n terms)
with the natural action of Σn . Then S 0 is a commutative monoid in S∗Σ . The multiplication
map
m
S0 ⊗ S0 → S0
comes from the natural maps
(Σp+q )+
^
S p ∧ S q → S p+q
Σp ×Σq
induced by the action of Σp+q on S p ∧ S q ∼
= S p+q . The unit morphism
0I Σ
S∗
= (S 0 , ∗, ∗, ∗, . . .) → (S 0 , S 1 , S 2 , . . .) = S 0
is the obvious one. The fact that S 0 is a commutative monoid is expressed by the commutativity of the diagram
S 0 ⊗ SH0
braiding
HH
HH
H
m HH
H$
S0
/ S0 ⊗ S0
vv
vv
v
vv m
vz v
which can be easily checked by hand.
2) If X ∈ S∗Σ , the additional structure of a symmetric spectrum on X is equivalent to
the additional structure of a left S 0 -module on X. We sketch an explanation and leave
the details to the reader as a very instructive exercise. First of all, a left S 0 -module on X
is a morphism S 0 ⊗ X → X satisfying a certain condition. This is the same as giving a
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morphism S 0 → HomS∗Σ (X, X) of monoids in S∗Σ (the inner hom on the right can be easily
made into a commutative monoid). Since S 0 is clearly “generated by S 1 as a monoid” in
a suitable sense, such a morphism of monoids is determined by “where S 1 goes”. Now it
is not hard to check that to specify where S 1 goes in a way that extends to all of S 0 is
equivalent to equipping X with a symmetric spectrum structure.
Once again, for all the details we refer the reader to [HSS], § 2.2.
We finally come to the key construction of the section. Note that if X ∈ S∗Σ is a left
0
S -module, then, because S 0 is commutative, we can turn X into a right S 0 -module using
the composition
braiding
action
X ⊗ S 0 −−−−→ S 0 ⊗ X −−−→ X.
Now if X and Y are symmetric spectra, we look at the corresponding right S 0 -module
structure ρX : X ⊗ S 0 → X and left S 0 -module structure λY : S 0 ⊗ Y → Y .
Definition: The smash product or tensor product of X and Y is defined by
ρX ⊗idY
/
/X ⊗Y
X ∧ Y = X ⊗ Y = Coequalizer X ⊗ S 0 ⊗ Y
.
idX ⊗λY
S0
The category S∗Σ is obviously complete and cocomplete, so X ⊗ Y is well defined, and,
S0
moreover, inherits a left action of S 0 , and hence becomes a symmetric spectrum.
Analogy: If k is a field, S is a commutative k-algebra, and M , N are S-modules, one
can construct their tensor product over S as
/
/ M ⊗N
M ⊗ N = Coeq M ⊗ S ⊗ N
.
S
k
k
k
Theorem ([HSS], Theorem 2.2.10): The smash product is a closed symmetric
monoidal structure on SpΣ .
We omit the proof and end with the following important
Definition: A symmetric ring spectrum is a monoid in SpΣ .
As far as I understand, the possibility of introducing this definition was one of the
original motivation behind the construction of the category SpΣ .
Post-lecture footnotes
As Drinfeld explained during the talk, there is a somewhat different point of view
on the notion of geometric realization. Namely, one can think of a simplicial set X as a
right 4
4-module. On the other hand, we have a natural left 4
4-module, which is the
functor 4 : 4
4 → Top obtained by sending [n] to the standard n-simplex |4[n]|. Then,
with the appropriate understanding of the tensor product, we have
∗)
|X| = X ⊗ 4 ∈ Top.
4
4
∗∗)
As remarked by Peter May during the lecture, the theory of orthogonal spectra
is no more complicated than the theory of symmetric spectra, and, in fact, it is more
convenient for certain purposes (for example, it is the case that a morphism of orthogonal
spectra is a stable homotopy equivalence if and only if it induces an isomorphism on all
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the stable homotopy groups). However, we have made a choice to work with symmetric
spectra throughout the minicourse; moreover, as far as I understand, orthogonal spectra
are more of a topological theory and it would be much harder to phrase it in the language
of simplicial sets.
Final comment. All of the second half of the lecture (i.e., starting with § 5) is
very formal in the sense that the category of simplicial sets could be replaced with any
closed symmetric monoidal category (e.g., that of vector spaces over a field), and all the
constructions and results would go through just as well.
Another remark is that to get a topological theory one needs to take this closed
symmetric monoidal category to be some convenient category of “spaces” , but here the
word “space” could also have several different meanings. For instance, we can look at the
categories of
• simplicial sets;
• weakly Hausdorff compactly generated topological spaces;
• topological spaces having the homotopy type of a CW complex.
It is possible to develop a theory of spectra and symmetric spectra in each of these
contexts. The main reason we choose to work with simplicial sets is that these are the
objects that arise most naturally in the applications to Beilinson’s work. However, it is
not necessary to work in S∗ to understand this lecture.
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